It is remarkable that Corollary REF  is even stronger than the main results of [1]}, [2]}
on the \(p\) -part of the BSD formula for semi-stable elliptic curves of analytic rank \(\le 1\) 
since the structure of Selmer groups is completely determined.
When \(\mathrm {ord}_{s=1}L(E,s) \le 1\) , we use Perrin-Riou's conjecture [3]} recently settled in [4]}, [5]}, [6]} instead of the Iwasawa main conjecture.
All the known results on the \(p\) -part of the BSD formula depend heavily on the full Iwasawa main conjecture (as listed in \(§\)REF ).
